Generators of Nonassociative Simple Moufang Loops over Finite Prime Fields

Transcription

1 Generators of Nonassociative Simple Moufang Loops over Finite Prime Fields Petr Vojtěchovský Department of Mathematics Iowa State University Ames IA USA We present an elementary proof that the nonassociative simple Moufang loops over finite prime fields are generated by three elements In the last section we conclude that integral Cayley numbers of unit norm are generated multiplicatively by three elements Key Words: simple Moufang loops economical generators integral Cayley numbers octonions 1 SIMPLE MOUFANG LOOPS The first class of nonassociative simple Moufang loops was discovered by L Paige in 1956 [9] who investigated Zorn s and Albert s construction of simple alternative rings M Liebeck proved in 1987 [7] that there are no other finite nonassociative simple Moufang loops We can briefly describe the class as follows: For every finite field F there is exactly one simple Moufang loop Recall Zorn s multiplication a α β b c γ = δ d ac + α δ βc + bδ + α γ aγ + αd β δ β γ + bd where a b c d F α β γ δ F 3 and where α β resp α β denotes the dot product resp vector product of α and β Probably the easiest way to think of this multiplication is to consider the usual matrix multiplication with the added antidiagonal matrix 0 β δ α γ 0 1

2 2 PETR VOJTĚCHOVSKÝ We define the determinant of a α M = β b by det M = ab α β Then L = {M; det M 0} turns out to be a nonassociative Moufang loop and so does M = {M; det M = 1} One can show that ZL the center of L consists of all elements a 0 0 a for 0 a F Thus ZM is at most a two-element subloop of M More precisely ZM is trivial if and only if char F the characteristic of F equals 2 cf Lemma 32 [9] Finally M/ZM was found to be simple and nonassociative in [9] Obviously a finite simple Moufang loop is either associative whence a finite simple group or nonassociative an element of the class introduced above Each finite simple group is known to be generated by just two elements See [11] for more details The proof of this fact depends heavily on the classification of finite simple groups As we have seen nonassociative simple Moufang loops admit a much simpler classification They cannot be generated by two elements only since every Moufang loop is diassociative cf [8] [2] or [10] In this paper we show that M/ZM is generated by three elements when F is a finite prime field The approach taken here is elementary Possible generalizations of this result would probably require more detailed methods than those used here or a completely different approach cf L E Dickson s proof that SL 2 q is two-generated for odd q 9 [4] or [5] 2 GENERATORS It is inconvenient to work with the quotient M/ZM We shall readily identify the elements M and M in all of our computations and multiply by 1 freely For M M a matrix let M denote the transpose of M In order to linearize our notation we introduce two mappings u l : F 3 M defined by uα = 1 α 1 0 lα = 0 1 α 1

3 GENERATORS OF SIMPLE MOUFANG LOOPS 3 Next consider t : F 3 \ {0} F 3 given by α if α 1 0 tα 1 α 2 α 3 = 0 α2 1 0 if α 1 = 0 α α3 1 otherwise Note that α tα = 1 Finally let sα stand for the matrix 0 α tα 0 Recall that for matrices in M 1 a α b α = β b β a and also the two special cases of [9]: lα = sα utα sα 1 uα = sαltα sα 2 Observe that lα 1 = l α and uα 1 = u α We start our search for generators with the following result due to Paige: Proposition 21 by Every simple Moufang loop M/ZM is generated {uα lα; 0 α F 3 } Proof Combine Lemmas 42 and 43 [9] Let us identify all nonzero elements of PF 3 = F 3 /F with those vectors from F 3 whose first nonzero coordinate equals 1 Also let e 1 = e 2 = and e 3 = as usual Proposition 22 generated by Assume that F is a prime field Then M/ZM is {uα lα; α PF 3 } Proof First check that uaαubα = ua + bα and laαlbα = la + bα for all α F 3 a b F Given 0 β F 3 there is a F and α PF 3 such that β = aα Since F is prime we can use a as an exponent and write uβ = uα a lβ = lα a We are finished by Proposition 21

4 4 PETR VOJTĚCHOVSKÝ Proposition 23 Assume that F is prime Then M/ZM is generated by S = {ue 1 ue 2 ue 3 } {sα sα ; α PF 3 } Proof Observe that for α PF 3 we have tα { e 1 e 2 e 3 } Given lα with α PF 3 1 yields lα = sα u e i sα for some e i 1 i 3 In particular the elements le i 1 i 3 are generated by S Symmetrically given uα with α PF 3 equation 2 yields uα = sαl e i sα for some e i 1 i 3; and we are done by Proposition 22 Lemma 21 Assume α β F 3 \ {0} are such that tα = tβ Then sβ = sαl α β and sβ = sα uα β Proof By definition we have α tβ tα tβ sαsβ = = α β tα β 1 0 α β 1 Thus sβ = sα 1 l α β = sαl α β As for the remaining equation start with sα sβ = uα β Proposition 24 Assume F is prime Then M/ZM is generated by {ue i se i ; 1 i 3} Proof Thanks to Proposition 23 we only need to generate elements sα and sα for α PF 3 First of all note that se i = se i and by 1 le i = se i ue i 1 se i for 1 i 3 Also s0 1 a = se 2 le 1 a and s0 1 a = se 2 ue 1 a for all a F Similarly s1 a 0 = se 1 le 3 a and s1 a 0 = se 1 ue 3 a for all a F Next by 1 and 2 l1 a 0 = s1 a 0 u e 1 s1 a 0 u1 a 0 = s1 a 0l e 1 s1 a 0 For 0 a F we have la = l1 a 0 a 1 ua = u1 a 0 a 1 Hence we obtained l a 1 0 and u a 1 0 for all a F

5 GENERATORS OF SIMPLE MOUFANG LOOPS 5 Finally Lemma 21 yields s1 a b = s1 a 0lb a 1 0 = s1 a 0l a 1 0 b s1 a b = s1 a 0 uba 1 0 = s1 a 0 u a 1 0 b for all b F We may further reduce the number of generators one by one down to three The equations 3 below have been carefully chosen They are satisfied independently of char F and are as simple as the author was able to find We leave the somewhat lengthy verification to the reader Let Then x = 0 e3 e 3 1 se 1 = se 3 se 2 se 2 = [ue 1 ue 3 ue 2 ue 1 ][ue 3 1 ue 2 se 3 ] ue 3 = x 1 se 3 se 3 = [ue 2 ue 1 x][xue 1 ] [x 2 ue 2 ue 1 x 2 ue 1 ] 3 Theorem 21 Assume that F is a prime field Then the simple Moufang loop M/ZM is generated by the three elements INTEGRAL CAYLEY NUMBERS OF UNIT NORM The smallest nonassociative simple Moufang loop M 120 of 120 elements is constructed over the binary field As Paige showed in [9] it is isomorphic to the integral Cayley numbers of unit norm modulo their center The aim of this short section is to offer one explicit isomorphism and to conclude that the loop of integral Cayley numbers of unit norm is generated multiplicatively by three elements The isomorphism will allow us to perform calculations inside integral Cayley numbers more efficiently than by the conventional rules Unlike the case of complex numbers C and quaternions H there are still many names for the eight-dimensional real algebra O = H H obtained from H by the Cayley-Dickson process: Cayley numbers algebra of octaves

6 6 PETR VOJTĚCHOVSKÝ octonions We prefer to use the name octonions Recall that for q Q r R H H the multiplication in O is defined by q Q r R = qr RQ Rq + Qr where the bar indicates conjugation in H An alternative way to describe the multiplication is to introduce a new unit e regard O as H + He and write q + Qe r + Re = qr RQ + Rq + Qre For an excellent discussion concerning the notation in O see [3] When O is viewed as an eight-dimensional real vector space we will use Dickson s notation for its basis namely 1 i j k e ie je and ke This is not the best choice when one wishes to describe the multiplicative relations between basis elements in a compact way see [1] [3] but it seems to be the best choice for what follows The conjugate of the octonion a = q + Qe is defined as a = q Qe Its norm Na is then the non-negative real number aa For any F = R C H O a set of integral elements of F is defined as a maximal subset of F containing 1 closed under multiplication and subtraction and such that both Na and a + a are integers for each element a of the set Such a set is unique for F = R C and H In the case of the octonions there are seven such sets all isomorphic For the rest of this paper we select the one which Coxeter denotes by J and calls integral Cayley numbers By J we mean the 240 elements of J with norm one See [3] p29 for the list of all elements of J No three of the basis elements generate J Following common practice let h = 1/2 i + j + k + e J Coxeter knew that i j and h generated J by multiplication and subtraction but he did not notice that J is generated by the above elements just under multiplication From 21 we know that there must be three elements generating J /{1 1} Indeed i j and h do the job One of the possible isomorphisms ϕ : J /{1 1} M 120 is determined by i 0 e3 e 3 0 j 0 e2 e 2 0 h which however is rather tedious to check by hand The author acknowledges that he used his own set of GAP 4 libraries [6] to confirm the computation Since i 2 = 1 we see that i j and h generate J In order to be able to carry out calculations in J we would like to know ϕe Lemma 31 i e = jh hi kh in O ii The element corresponding to e under ϕ is

7 GENERATORS OF SIMPLE MOUFANG LOOPS 7 Proof Consider the multiplicative relations in [3] p567 or p28 In particular we have hi = 1 ih Hence jh hi kh = jh + jh ih kh = jh + k h ih kh = jh kh + k kh h kh ih kh = i + h kh + h k h j h kh = i j k + 2h which equals e since h = 1/2 i + j + k + e REFERENCES 1 J H Conway and al ATLAS of Finite Groups Oxford University Press New York R H Bruck A Survey of Binary Systems Ergebnisse der Mathematik und ihrer Grenzgebiete Neue Folge Heft 20 Springer-Verlag H S M Coxeter Integral Cayley Numbers Duke Mathematical Journal Vol 13 No 4 December 1946 Reprinted in H S M Coxeter Twelve Geometric Essays Southern Illinois University Press L E Dickson Linear Groups with an Exposition of the Galois Field Theory Teubner 1901; reprinted by Dover S Doro Simple Moufang loops Math Proc Camb Phil Soc pp The GAP Group GAP Groups Algorithms and Programming Version 41; Aachen St Andrews gap The specific libraries related to this paper are available at petr 7 M W Liebeck The classification of finite simple Moufang loops Math Proc Camb Phil Soc pp R Moufang Zur Struktur von Alternativkörpern Math Ann pp L Paige A Class of Simple Moufang Loops Proceedings of the American Mathematical Society Vol 7 Issue 3 pp June H O Pflugfelder Quasigroups and Loops: Introduction Sigma series in pure mathematics; 7 Heldermann Verlag Berlin J S Wilson Economical Generating Sets for Finite Simple Groups London Mathematical Society Lecture Note Series 207 Groups of Lie Type and Their Geometries Cambridge University Press 1995